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5.2 Stress-energy bi-tensor operator and noise kernel

Even though we believe that the point-separated results are more basic in the sense that it reflects a deeper structure of the quantum theory of spacetime, we will nevertheless start with quantities defined at one point, because they are what enter in conventional quantum field theory. We will use point separation to introduce the bi-quantities. The key issue here is thus the distinction between point-defined (pt) and point-separated (bi) quantities.

For a free classical scalar field φ with the action Sm [g,φ] defined in Equation (1View Equation), the classical stress-energy tensor is

( ) 1- ;p p Tab = (1 − 2ξ)φ;aφ;b + 2ξ − 2 φ;pφ gab + 2ξφ (φ;p − φ;abgab) ( ) + φ2ξ Rab − 1R gab − 1m2 φ2gab, (49 ) 2 2
which is equivalent to the tensor of Equation (3View Equation), but written in a slightly different form for convenience. When we make the transition to quantum field theory, we promote the field φ(x ) to a field operator φˆ(x ). The fundamental problem of defining a quantum operator for the stress tensor is immediately visible: The field operator appears quadratically. Since ˆφ(x ) is an operator-valued distribution, products at a single point are not well-defined. But if the product is point separated, ˆφ2(x) → φˆ(x)ˆφ(x′), they are finite and well-defined.

Let us first seek a point-separated extension of these classical quantities and then consider the quantum field operators. Point separation is symmetrically extended to products of covariant derivatives of the field according to

1( ′ ′ ) (φ;a) (φ;b) → -- gap∇p ′∇b + gbp∇a ∇p ′ φ (x )φ(x′), 2( ) φ (φ ) → 1- ∇ ∇ + g p′gq′∇ ′∇ ′ φ (x )φ(x′). ;ab 2 a b a b p q
The bi-vector of parallel displacement g a′(x, x′) a is included so that we may have objects that are rank 2 tensors at x and scalars at ′ x.

To carry out point separation on Equation (49View Equation), we first define the differential operator

( ) 1 ( a′ b′ ) 1 cd′ 𝒯ab = --(1 − 2ξ) ga ∇a ′∇b + gb ∇a ∇b ′ + 2ξ − -- gabg ∇c∇d ′ 2 ( ) ( 2 ) ( ) a′ b′ c c′ 1- 1- 2 − ξ ∇a ∇b + ga gb ∇a ′∇b′ + ξgab ∇c ∇ + ∇c ′∇ + ξ Rab − 2gabR − 2m gab, (50 )
from which we obtain the classical stress tensor as
Tab(x) = lim 𝒯abφ (x )φ(x′). (51 ) x′→x
That the classical tensor field no longer appears as a product of scalar fields at a single point allows a smooth transition to the quantum tensor field. From the viewpoint of the stress tensor, the separation of points is an artificial construct, so when promoting the classical field to a quantum one, neither point should be favored. The product of field configurations is taken to be the symmetrized operator product, denoted by curly brackets:
1 { } 1( ) φ(x)φ(y) → -- φˆ(x ), ˆφ(y) = -- ˆφ(x)ˆφ(y) + ˆφ(y)φˆ(x ) (52 ) 2 2
With this, the point separated stress-energy tensor operator is defined as
{ } ˆ ′ 1- ˆ ˆ ′ Tab(x, x) ≡ 2𝒯ab φ(x),φ(x ) . (53 )
While the classical stress tensor was defined at the coincidence limit x′ → x, we cannot attach any physical meaning to the quantum stress tensor at one point until the issue of regularization is dealt with, which will happen in the next section. For now, we will maintain point separation so as to have a mathematically meaningful operator.

The expectation value of the point-separated stress tensor can now be taken. This amounts to replacing the field operators by their expectation value, which is given by the Hadamard (or Schwinger) function

⟨{ }⟩ G (1)(x,x′) = ˆφ(x), ˆφ(x′) , (54 )
and the point-separated stress tensor is defined as
ˆ ′ 1- (1) ′ ⟨Tab(x,x )⟩ = 2 𝒯abG (x,x ), (55 )
where, since 𝒯 ab is a differential operator, it can be taken “outside” the expectation value. The expectation value of the point-separated quantum stress tensor for a free, massless (m = 0) conformally coupled (ξ = 16) scalar field on a four dimension spacetime with scalar curvature R is
( ) ⟨ˆTab(x,x′)⟩ = 1- gp′bG (1);p′a + gp′aG (1);p′b − -1gp′q G (1);p′q gab 6 ( ) 12 [( ) ] 1-- p′ q′ (1) ′′ (1) 1-- (1) ′p′ (1) p − 12 g a g b G ;pq + G ;ab + 12 G ;p + G ;p gab 1 ( 1 ) + --G (1) Rab − --R gab . (56 ) 12 2

5.2.1 Finiteness of the noise kernel

We now turn our attention to the noise kernel introduced in Equation (11View Equation), which is the symmetrized product of the (mean subtracted) stress tensor operator:

⟨ { ⟨ ⟩ ⟨ ⟩} ⟩ 8Nab,c′d′(x,y) = Tˆab(x) − Tˆab(x) ,Tˆc′d′(y) − Tˆc′d′(y) ⟨ { } ⟩ ⟨ ⟩⟨ ⟩ = Tˆab(x),Tˆc′d′(y) − 2 ˆTab(x ) ˆTc′d′(y) . (57 )
Since Tˆab(x) defined at one point can be ill-behaved as it is generally divergent, one can question the soundness of these quantities. But as will be shown later, the noise kernel is finite for y ⁄= x. All field operator products present in the first expectation value that could be divergent, are canceled by similar products in the second term. We will replace each of the stress tensor operators in the above expression for the noise kernel by their point separated versions, effectively separating the two points (x,y ) into the four points (x,x′,y,y′). This will allow us to express the noise kernel in terms of a pair of differential operators acting on a combination of four and two point functions. Wick’s theorem will allow the four point functions to be re-expressed in terms of two point functions. From this we see that all possible divergences for y ⁄= x will cancel. When the coincidence limit is taken, divergences do occur. The above procedure will allow us to isolate the divergences and to obtain a finite result.

Taking the point-separated quantities as more basic, one should replace each of the stress tensor operators in the above with the corresponding point separated version (53View Equation), with 𝒯ab acting at x and x′ and 𝒯c′d′ acting at y and y′. In this framework the noise kernel is defined as

8N ′′(x,y) = lim lim 𝒯 𝒯 ′′ G (x,x′,y,y′), (58 ) ab,c d x′→x y′→y ab cd
where the four point function is
′ ′ 1 [⟨ {{ ′ } { ′} }⟩ ⟨{ ′} ⟩⟨ { ′} ⟩] G(x, x,y, y) = -- ˆφ(x), ˆφ(x ) , φˆ(y ), ˆφ(y) − 2 ˆφ(x),φˆ(x ) φˆ(y), ˆφ(y ) .(59 ) 4
We assume that the pairs ′ (x,x ) and ′ (y,y ) are each within their respective Riemann normal coordinate neighborhoods so as to avoid problems that possible geodesic caustics might be present. When we later turn our attention to computing the limit y → x, after issues of regularization are addressed, we will want to assume that all four points are within the same Riemann normal coordinate neighborhood.

Wick’s theorem, for the case of free fields which we are considering, gives the simple product four point function in terms of a sum of products of Wightman functions (we use the shorthand notation G ≡ G (x,y) = ⟨ˆφ(x ) ˆφ(y)⟩ xy +):

⟨ ⟩ ˆφ(x ) ˆφ(y)φˆ(x′) ˆφ(y′) = G ′ G ′ + G ′ G ′ + G G ′ ′ (60 ) xy yx xx yy xy x y
Expanding out the anti-commutators in Equation (59View Equation) and applying Wick’s theorem, the four point function becomes
′ ′ G(x,x ,y,y ) = Gxy ′ Gx′y + Gxy Gx ′y′ + Gyx′ Gy ′x + Gyx Gy ′x′. (61 )
We can now easily see that the noise kernel defined via this function is indeed well defined for the limit (x′,y′) → (x,y):
( 2 2) G(x,x, y,y) = 2 G xy + G yx . (62 )
From this we can see that the noise kernel is also well defined for y ⁄= x; any divergence present in the first expectation value of Equation (59View Equation) have been cancelled by those present in the pair of Green functions in the second term, in agreement with the results of Section 3.

5.2.2 Explicit form of the noise kernel

We will let the points separated for a while so we can keep track of which covariant derivative acts on which arguments of which Wightman function. As an example (the complete calculation is quite long), consider the result of the first set of covariant derivative operators in the differential operator (50View Equation), from both 𝒯ab and 𝒯c′d′, acting on ′ ′ G (x,x ,y,y ):

1- 2 ( p′′ p′′ )( q′′′ q′′′ ) ′ ′ 4 (1 − 2 ξ) ga ∇p ′′∇b + gb ∇p ′′∇a gc′ ∇q ′′′∇d ′ + gd′ ∇q ′′′∇c ′ G (x,x ,y,y ). (63 )
(Our notation is that ∇a acts at x, ∇c′ at y, ∇b ′′ at ′ x, and ∇d ′′′ at ′ y.) Expanding out the differential operator above, we can determine which derivatives act on which Wightman function:

2 [ (1-−-2-ξ)-× g′p′′′gq′′ (G ′ ′′′ G ′ ′′′ + G ′ G ′′ ′′ ′′′ 4 c a xy ;bp x y;q d xy;bd xy ;q p + Gyx′;q′′d′ Gy ′x;bp′′′ + Gyx;bd′ Gy′x′;q′′p′′′) ′′′ ′′ +gd ′p gq a(Gxy ′;bp′′′ Gx ′y;q′′c′ + Gxy;bc′ Gx′y′;q′′p′′′ + Gyx′;q′′c′ Gy′x;bp′′′ + Gyx;bc′ Gy ′x′;q′′p′′′) +g ′p′′′gq′′ (G ′ ′′′ G ′ ′′ ′ + G ′ G ′′ ′′′′′ c b xy ;ap xy;q d xy;ad xy;q p + Gyx′;q′′d′ Gy ′x;ap′′′ + Gyx;ad′ Gy ′x′;q′′p′′′) p′′′ q′′ +gd ′ g b(Gxy′;ap′′′ Gx ′y;q′′c′ + Gxy;ac′ Gx′y′;q′′p′′′ ] + Gyx′;q′′c′ Gy′x;ap′′′ + Gyx;ac′ Gy′x′;q′′p′′′) . (64 )
If we now let ′ x → x and ′ y → y, the contribution to the noise kernel is (including the factor of 1 8 present in the definition of the noise kernel):
1[(1 − 2ξ)2(G ′G ′ + G ′G ′) + (1 − 2 ξ)2(G ′G ′ + G ′G ′)]. (65 ) 8 xy;ad xy;bc xy;ac xy;bd yx;ad yx;bc yx;ac yx;bd
That this term can be written as the sum of a part involving Gxy and one involving Gyx is a general property of the entire noise kernel. It thus takes the form
Nabc′d′(x,y) = Nabc′d′ [G+ (x, y)] + Nabc′d′ [G+ (y,x)]. (66 )
We will present the form of the functional Nabc′d′ [G ] shortly. First we note, that for x and y time-like separated, the above split of the noise kernel allows us to express it in terms of the Feynman (time ordered) Green function GF (x, y) and the Dyson (anti-time ordered) Green function GD (x,y):
Nabc′d′(x,y ) = Nabc′d′ [GF (x,y )] + Nabc′d′ [GD (x,y)]. (67 )
This can be connected with the zeta function approach to this problem [241] as follows: Recall when the quantum stress tensor fluctuations determined in the Euclidean section is analytically continued back to Lorentzian signature (τ → it), the time ordered product results. On the other hand, if the continuation is τ → − it, the anti-time ordered product results. With this in mind, the noise kernel is seen to be related to the quantum stress tensor fluctuations derived via the effective action as
| | 16Nabc′d′ = ΔT a2bc′d′|t=− iτ,t′= −iτ′ + ΔT 2abc′d′|t=iτ,t′=iτ′ . (68 )
The complete form of the functional Nabc ′d′ [G ] is
Nabc′d′ [G ] = N&tidle;abc′d′ [G ] + gabN&tidle;c ′d′ [G] + gc′d′N &tidle;′ [G ] + gabgc′d′N &tidle; [G ], (69 ) ab
with
8N&tidle; ′′ [G ] = (1 − 2ξ)2 (G ′ G ′ + G ′ G ′) + 4ξ2 (G ′ ′ G + G G ′′) abc d ;cb ;d a ;ca ;d b ;cd ;ab ;abcd − 2ξ (1 − 2ξ)(G;bG;c′ad′ + G;a G;c′bd′ + G;d′ G;abc′ + G;c′ G;abd′) +2 ξ (1 − 2ξ)(G G R ′ ′ + G ′ G ′ R ) 2 ;a ;b cd ;c ;d2 ab 2 − 4ξ (G;abRc′d′ + G;c′d′ Rab )G + 2ξ Rc ′d′ RabG , (70 ) [( ) ] ′ 1 p′ ( p′ p′) 8 &tidle;Nab[G ] = 2 (1 − 2ξ ) 2ξ − -- G;p′bG; a + ξ G;b G;p′a + G;a G;p′b [( ) 2 ] 1- p′ ( p′ p′) − 4ξ 2ξ − 2 G; G;abp′ + ξ G;p′ G;ab + G G;abp′ 2 ′ − (m [+( ξR )[()1 − 2ξ)G;aG;b − 2G ξ G;]ab] 1 p′ p′ 2 ′ 2 +2 ξ 2ξ − -- G;p′ G; + 2G ξ G;p′ Rab − (m + ξR )ξ Rab G , (71 ) 2 ( ) &tidle; 1- 2 p′q 2( p′ q p q′) 8N [G ] = 2 2ξ − 2 G;p′q G; + 4ξ G;p′ G;q + G G;p q′ ( ) ( ) +4 ξ 2 ξ − 1- G G ′pq′ + G p′ G q ′ 2 ;p ;q ; ;q p ( 1) [( ) ( ) ] − 2ξ − -- m2 + ξR G;p′ G;p′ + m2 + ξR ′ G;pG;p [ 2 ] ( 2 ) p′ ( 2 ′) p − 2ξ m + ξR G;p′ + m + ξR G;p G 1 ( 2 )( 2 ′) 2 + -- m + ξR m + ξR G . (72 ) 2

5.2.3 Trace of the noise kernel

One of the most interesting and surprising results to come out of the investigations of the quantum stress tensor undertaken in the 1970s was the discovery of the trace anomaly [6184]. When the trace of the stress tensor T = gabTab is evaluated for a field configuration that satisties the field equation (2View Equation), the trace is seen to vanish for massless conformally coupled fields. When this analysis is carried over to the renormalized expectation value of the quantum stress tensor, the trace no longer vanishes. Wald [283] showed that this was due to the failure of the renormalized Hadamard function Gren(x,x′) to be symmetric in x and x ′, implying that it does not necessarily satisfy the field equation (2View Equation) in the variable x′. (The definition of Gren(x,x ′) in the context of point separation will come next.)

With this in mind, we can now determine the noise associated with the trace. Taking the trace at both points x and y of the noise kernel functional (67View Equation) yields

ab c′d′ N [G ] = g g N[a(bc′d′[G] ) ( ) ] 2 1- p′ 2 1- ′ p = − 3 G ξ m + 2 ξR G;p′ + m + 2ξR G;p 2( ) ( ) ( ) + 9ξ-- G ′p′ G p + G G p ′p′ + m2 + 1ξR m2 + 1-ξR ′ G2 2 ;p ;p ;p p 2 2 ( 1 ) [ ( 1 ) ′ ( ′ ′) +3 --− ξ 3 --− ξ G;p′pG;p p − 3ξ G;p G;p′pp + G;p′ G;ppp 6 ( 6 ) ( ) ] 2 1 p′ 2 1 ′ ;p + m + -ξR G;p′ G; + m + --ξR G;p G . (73 ) 2 2
For the massless conformal case, this reduces to
N [G ] = -1--{RR ′G2 − 6G (R □′ + R′□ )G + 18[(□G )(□ ′G) + □ ′□G ]}, (74 ) 144
which holds for any function G(x, y). For G being the Green function, it satisfies the field equation (2View Equation):
2 □G = (m + ξR)G. (75 )
We will only assume that the Green function satisfies the field equation in its first variable. Using the fact □ ′R = 0 (because the covariant derivatives act at a different point than at which R is supported), it follows that
□ ′□G = (m2 + ξR)□ ′G. (76 )
With these results, the noise kernel trace becomes
1 [ ( 1 ) ][ ( ) ] N [G ] = -- m2 (1 − 3ξ) + 3R --− ξ ξ G2 2m2 + R ′ξ + (1 − 6 ξ)G;p′ G;p′ − 6G ξG;p′p′ 2 ( ) [ 6 ( ) ] 1 1 ( 2 ′ ) ;p pp′ 1 p′p + -- --− ξ 3 2m + R ξ G;p G − 18ξ G;pG;p′ + 18 --− ξ G;p′p G; , (77 ) 2 6 6
which vanishes for the massless conformal case. We have thus shown, based solely on the definition of the point separated noise kernel, that there is no noise associated with the trace anomaly. This result obtained in [245Jump To The Next Citation Point] is completely general since it is assumed that the Green function is only satisfying the field equations in its first variable; an alternative proof of this result was given in [208]. This condition holds not just for the classical field case, but also for the regularized quantum case, where one does not expect the Green function to satisfy the field equation in both variables. One can see this result from the simple observation used in Section 3: Since the trace anomaly is known to be locally determined and quantum state independent, whereas the noise present in the quantum field is non-local, it is hard to find a noise associated with it. This general result is in agreement with previous findings [44Jump To The Next Citation Point167Jump To The Next Citation Point58Jump To The Next Citation Point], derived from the Feynman-Vernon influence functional formalism [8988] for some particular cases.


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